HEN OF MATHEMATICS
line r through P such that V never meets Z no matter how far /'
and I are extended (in either direction). Merely as a nominal
definition -we say that two straight lines lying in one plane -
which never meet are parallel Thus the fifth postulate of Euclid
asserts that through P there is precisely one straight line
parallel to L Euclid's penetrating insight into the nature of
geometry convinced him that this postulate had not, in his
time, been deduced from the others, although there had been
many attempts to prove the postulate. Being unable to deduce
the postulate himself from his other assumptions, and wishing
to use it in the proofs of many of his theorems, Euclid honestly
set it out with his other postulates.
There are one or two simple matters to be disposed of before
we come to Lobatchewsky's Copernican part in the extension
of geometry. We have alluded to 'equivalents' of the parallel
postulate. One of these, 'the hypothesis of the right angle', as it
is called, will suggest two possibilities, neither equivalent to
Euclid's assumption, one of which introduces Lobatchewsky's
geometry, the other, RiemamVs.
Y
Consider a figure AXYB which 'looks like' a rectangle, con-
sisting of four straight lines AX, XY9 YJ3, BA, in which BA
(or AB) is the base, AX and YB (or BY) are drawn equal and
perpendicular to AB, and on the same side of AB. The essential
things to be remembered about this figure are that each of the
angles XAB> YBA (at the base) is a right angle, and that the
sides „«, BY are equal in length. Without using the parallel
postulate, it can be proved that the angles AXY, *BYX, are
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